4,180 research outputs found
Trees and Markov convexity
We show that an infinite weighted tree admits a bi-Lipschitz embedding into
Hilbert space if and only if it does not contain arbitrarily large complete
binary trees with uniformly bounded distortion. We also introduce a new metric
invariant called Markov convexity, and show how it can be used to compute the
Euclidean distortion of any metric tree up to universal factors
Fat Polygonal Partitions with Applications to Visualization and Embeddings
Let be a rooted and weighted tree, where the weight of any node
is equal to the sum of the weights of its children. The popular Treemap
algorithm visualizes such a tree as a hierarchical partition of a square into
rectangles, where the area of the rectangle corresponding to any node in
is equal to the weight of that node. The aspect ratio of the
rectangles in such a rectangular partition necessarily depends on the weights
and can become arbitrarily high.
We introduce a new hierarchical partition scheme, called a polygonal
partition, which uses convex polygons rather than just rectangles. We present
two methods for constructing polygonal partitions, both having guarantees on
the worst-case aspect ratio of the constructed polygons; in particular, both
methods guarantee a bound on the aspect ratio that is independent of the
weights of the nodes.
We also consider rectangular partitions with slack, where the areas of the
rectangles may differ slightly from the weights of the corresponding nodes. We
show that this makes it possible to obtain partitions with constant aspect
ratio. This result generalizes to hyper-rectangular partitions in
. We use these partitions with slack for embedding ultrametrics
into -dimensional Euclidean space: we give a -approximation algorithm for embedding -point ultrametrics
into with minimum distortion, where denotes the spread
of the metric, i.e., the ratio between the largest and the smallest distance
between two points. The previously best-known approximation ratio for this
problem was polynomial in . This is the first algorithm for embedding a
non-trivial family of weighted-graph metrics into a space of constant dimension
that achieves polylogarithmic approximation ratio.Comment: 26 page
Markov convexity and nonembeddability of the Heisenberg group
We compute the Markov convexity invariant of the continuous infinite
dimensional Heisenberg group to show that it is Markov
4-convex and cannot be Markov -convex for any . As Markov convexity
is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a
different proof of the classical theorem of Pansu and Semmes that the
Heisenberg group does not admit a biLipschitz embedding into any Euclidean
space.
The Markov convexity lower bound will follow from exhibiting an explicit
embedding of Laakso graphs into that has distortion
at most . We use this to show that if is a Markov
-convex metric space, then balls of the discrete Heisenberg group
of radius embed into with distortion at least
some constant multiple of
Finally, we show that Markov 4-convexity does not give the optimal distortion
for embeddings of binary trees into by showing that
the distortion is on the order of .Comment: version to appear in Ann. Inst. Fourie
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
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